Int. Statistical Inst.: Proc. 58th Worl Statistical Congress, 2011, Dublin (Session CPS039) p.6567 Estimation of District Level Poor Househols in the State of Uttar Praesh in Inia by Combining NSSO Survey an Census Data- An Application of Small Area Estimation Chanra, Hukum Inian Agricultural Statistics Research Institute Library Avenue, PUSA New Delhi-110012, Inia E-mail: hchanra@iasri.res.in Su, Umesh Inian Agricultural Statistics Research Institute Library Avenue, PUSA New Delhi-110012, Inia E-mail: ucsu@iasri.res.in Salvati, Nicola University of Pisa, Pisa, Italy E-mail:salvati@ec.unipi.it 1. INTRODUCTION There is great emphasis on istrict level planning in Inia. The efforts to evelop atabases require for planning an ecision-making at lower than the State level, were initiate quite some time back with the Planning Commission in the Government of Inia setting up a Working Group on Districts Planning in September, 1982. The Working Group in its report clearly highlighte the ata requirement for planning an ecision-making at the istrict level. However, it was foun that though a lot of ata are collecte, processe an publishe for the country as a whole or for iniviual states, not much isaggregation of the ata for substate level is one. The National Sample Survey Organisation (NSSO) surveys are main source of official statistics in Inia. However, these surveys are planne to generate statistics at state an national level. There is no regular flow of estimates at further below level, e.g., at the istrict level. Therefore, NSSO surveys provie reliable state an national level estimates; they can not be use to erive reliable irect estimates at the istrict level owing to small sample sizes which lea to high levels of sampling variability (see [7] an [8]). Although in Inian context istrict is a very important omain of planning process, we o not have surveys to prouce estimates at this level. At the same time it is also true that conucting istrict specific surveys is going to be very trivial an costly as well as time consuming job. Using the state level survey ata to erive the estimates at istrict or further smaller level may result in small sample sizes leaing to very unstable estimates. Due to the lack of statistics at lower level, proper planning, fun allocation an also monitoring of various plans is likely to suffer. An obvious solution to this problem is to use small area estimation (SAE) techniques. The SAE prouces reliable estimates for such small areas with small sample sizes by borrowing strength from ata of other areas. The SAE techniques are generally base on moel-base methos. The
Int. Statistical Inst.: Proc. 58th Worl Statistical Congress, 2011, Dublin (Session CPS039) p.6568 iea is to use statistical moels to link the variable of interest with auxiliary information, e.g. Census an Aministrative ata, for the small areas to efine moel-base estimators for these areas. Small area moels can be classifie into two broa types: (i) Area level ranom effect moels, which are use when auxiliary information is available only at area level. They relate small area irect estimates to area-specific covariates (Fay an Herriot [4]) an (ii) Neste error unit level regression moels, propose originally by Battese, Harter an Fuller [2]. These moels relate the unit values of a stuy variable to unit-specific covariates. We aopt the area level moel since covariates are available only at the area level. In this paper we use SAE techniques to erive moel-base estimates of proportion of poor househols at small area levels in the State of Uttar Praesh in Inia by linking ata from the Househol Consumer Expeniture (HCE) Survey 2006-07 of NSSO 63r roun an the Population Census 2001. Small areas are efine as the ifferent istricts of State of Uttar Praesh in Inia. The rest of the paper is organise as follows. In Section 2 we escribe the ata use for the analysis an in Section 3 we present an overview of the methoology use for analysis. Section 4 iscusses the iagnostic proceures for examining the moel assumptions, valiating the small area estimates an escribes the results. Section 5 finally set out the main conclusions. 2. DATA Two types of variables are require for this analysis. (1) The variable of interest for which small area estimates are require is rawn from the HCE Survey 2006-07 of NSSO 63r roun ata for rural areas of the State of Uttar Praesh. The target variable use for the stuy was poor househols. The poverty line has been use to ientify whether given househol is poor or not. A househol having monthly per capita consumer expeniture below the state s poverty line (i.e., Rs. 365.84) is categorise as poor househol. The poverty line use in this stuy is same as those of year 2004-05, given by planning commission, Govt of Inia. The parameter of interest is the proportion of poor househol at the istrict level. (2) The auxiliary (covariates) variables are rawn from the Population Census 2001. There were more than 100 covariates available for the purpose of moelling. Out of these, suitable covariates were selecte for the analysis as follows: We first examine the correlation of all these covariates with the target variable an then selecte the covariates with reasonably goo correlation with the target variable. This was followe by step-wise regression analysis. Finally, six variables namely (i) sex ratio of SC population, (ii) sex ratio of ST population, (iii) percentage of Other worker Population, (iv) percentage of Literate Male, (v) main Other workers female an (vi) marginal Other population were ientifie for the further analysis which significantly explaine the moel. The R 2 for the chosen moel was 48 per cent. The sampling esign use in the NSSO ata is stratifie multi-stage ranom sampling with istricts as strata, villages as first stage units an househols as the secon stage units. A total of 2322 househols were surveye from the 70 istricts of the Uttar Praesh. The istrict-wise sample size varie from 19 to 48 with average of 33 (Table 1). Our aim is to estimate proportion of poor househols at istrict level. It is evient that istrict level sample sizes are very small with very low values of average sampling fraction as 0.0001. Therefore, it is ifficult to erive reliable estimates an their stanar errors at istrict level. The SAE is an obvious choice for such cases. 3. METHODOLOGY To start with, we fix our notations. Throughout, we use a subscript to inex the quantities belonging to small area ( = 1,..., D), where D is the number of small areas (or areas) in the population. The subscript s
Int. Statistical Inst.: Proc. 58th Worl Statistical Congress, 2011, Dublin (Session CPS039) p.6569 an r are use for enoting the quantities relate to the sample an non-sample parts of the population. So that n an N represent the sample an population (i.e., number of househols in sample an population) sizes in istrict area is efine by, respectively. The value of variable of interest y (the poor househols) in the y an we enote by househols in area. Inee, the variable of interest an π, enote by ys ~ Bin( n, π ), where π y s an y r the sample an non-sample counts of poor y has a Binomial istribution with parameters terme as the probability of a success. Sim ilarly, yr ~ Bin( N n, π ). Further, y s an y r are assume to be inepenent Binomial variables with s is the probability of a poor househol in area, often π being a common success probability. As mentione in previous Section in moel-base small area estimation the survey ata is supplemente by the availability of auxiliary information from various sources, e.g., Census an Aministrative recors. Let x be the k- vector of the covariates for area from the previous sources. The moel linking the probabilities of success π with the covariates x is the logistic linear mixe moel given by where β { ( ) 1 } logit( π ln π 1 π η ) = = = xβ + u,( = 1,..., D ), is the k-vector of regression coefficient often known as fixe effect parameters an u is the area-specific ranom effect that accounts for between area issimilarity beyon that explaine by the auxiliary variables inclue in the fixe part of the moel. We assume that u s are inepenent an normally istribute with mean zero an variance ϕ. Uner moel (1), we get π exp( η ){ 1 exp( η )} 1 (1) n = +. It is evient that moel (1) relates the area level proportions to area level covariates. This type of moel is often referre to as area-level moel in SAE terminology, see for example [8]. Such a moel was originally use by Fay an Herriot [4] for the preiction of mean per-capita income (PCI) in small geographical areas (less than 500 persons) within counties in the Unite States. The Fay an Herriot (FH) metho for SAE is base on area level linear mixe moel an their approach is applicable to a continuous variable. In contrast, moel (1) is a special case of a generalize linear mixe moel (GLMM) with logit link function (see [3]) an suitable for iscrete, particularly binary variable. It is noteworthy that the FH moel is not applicable in such cases. Saei an Chambers [9] an Manteiga et al. [6] escribe this moel in the context of SAE. By efinition, the means of y an s y r given u uner moel (1) are: ( u ) π x β ( E y = n = exp( + u ) 1+ exp( + u )) 1 s n x β E( y u r ) = ( N n ) e u) ( 1 exp( )) 1 π = ( N n ) xp( x β + + x β + u. (3) Let T. We can write + enotes the total number of poor househols in istrict T = ys yr, where the first term y s, the sample count is known whereas the secon term y r, the non-sample count, is unknown. Therefore, an estimate T ˆ of the total number of poor househols in area is obtaine by replacing y r by its preicte value uner the moel (1). That is, ( + x ˆ ) 1 + u Tˆ y ˆ y ( N n) ex ˆ ˆ 1 exp( ˆ = s + yr = s + p( x β + u ) β ). (4) (2)
Int. Statistical Inst.: Proc. 58th Worl Statistical Congress, 2011, Dublin (Session CPS039) p.6570 An estimate of proportion of poor househols p in a small area is obtaine as 1 { ( ) exp( x ˆ )( 1 exp( x ˆ )) } ˆ 1 1 pˆ ˆ ˆ = TN = N ys + N n β + u + β + u. (5) It is obvious that in orer to compute the estimates given by equation (4) or (5), we require estimates of the unknown parameters β an u. A major ifficulty in use of logistic linear mixe m oel (LLMM) for SAE is the estimation of unknown moel parameters since the likelihoo function for LLMM often involves high imensional integrals (compute by integrating a prouct of iscrete an normal ensities, which has no analytical solution) which are ifficult to evaluate numerically. We use an iterative proceure that combines the Penalize Quasi-Likelihoo (PQL) estimation of β an u = ( u1,..., u D ) with REML estimation of φ to estimate these unknown parameters. Detaile escription of the approach can be followe from [6, 9]. We now turn to estimation of mean square error (MSE) for preictors given by equation (5). The MSE estimates are compute to assess the reliability of estimates an also to construct the confience interval (CI) for the estimates. The MSE estimate of (5) uner moel (1) is (see [6, 9]) given by mse( pˆ ) = m ( ˆ φ) + m ( ˆ φ) + 2 m ( ˆ φ). (6) 1 2 3 The first two components m 1 an m 2 constitute the largest part of the overall MSE estimates in (6). These are the MSE of the best linear unbiase preictor (BLUP)-type estimator when φ is known ([8]). The thir component m 3 is the variability ue to the estimate of φ. For simplicity, we use few notations to write the analytical expression of various components of the MSE (6). We enote by ˆ = iag{ n pˆ (1 pˆ )} {( ) (1 )} V an s V ˆ = iag N n pˆ pˆ, the iagonal matrices efine by the co rresponing variances of the r sample an non-sample part re spectively. = { iag( N )} 1 ˆ 1 A V r, B= { iag( N )} ˆ ˆ ˆ Vr r A svs s X T X an 1 ( φ ˆ ) 1 T ˆ s = ID + Vs, where X s an X are the sample an non-sample part of covariates an r I D is an ientity matrix of orer D. We further write Tˆ = { X Vˆ X X Vˆ TˆVˆ X } 1 an (1) s s s s s s s s Tˆ =Tˆ + TˆVˆ X Tˆ X Vˆ T ˆ. With these notations, assuming moel (1) hols, the various components (2) s s s s (1) s s s of equation (6) are m1 ( ˆ φ ) = AT ˆ sa, m2 ( ˆ φ ) ˆ ( 1), an = BT B m3 ( ˆ φ) = trace( ˆ ˆ ˆ ( ˆ i jv φ) ) Σ with Σ ˆ = Vˆ +φ ˆI Vˆ Vˆ. s D s s Here v( φ) ˆ is the asymptotic covariance matrix of the estimates of variance co mponents ˆ φ, which can be evaluate as the inverse of the appropriate Fisher information matrix for ˆ φ. Note that this also epens upon whether we are using maximum likelihoo (ML) or restricte maximum likelihoo ( REML) estimates for ˆ φ. We use REML estimates for ˆ φ, then v( ˆ φ 2 ( ˆ2 ( D 2 t ˆ 4 φ 1) + φ t11) 1 )= with t 1 = φˆ trace( T (2) ) 1 ˆ
Int. Statistical Inst.: Proc. 58th Worl Statistical Congress, 2011, Dublin (Session CPS039) p.6571 t = trace( Tˆ T ˆ ). Let us write an 11 (2) (2) is t he th i row of the matrix A. ˆ = ( Δ ) φ = ( ATˆ ) φ, where Δ=AT ˆs an i i φ= ˆ φ i s φ = ˆ φ A i 4. RESULTS AND DISCUSSIONS Generally two types of iagnostics proceures are teste in SAE, the moel iagnostics an the iagnostics for the small area estimates, see for example [1]. The first iagnostics are use to verify the assumptions of unerlying moel an the secon iagnostics are applie to valiate the reliability of the moel-base small area estimates. The ranom area effects u ( = 1,..., D) in moel (1) are assume to have a normal istribution with mean zero an variance ϕ. If the moel assumptions are satisfie then the istrict level resiuals are expecte to be ranomly istribute an not significantly ifferent from the regression line y=0, where uner moel (1), the area level resiuals are efine as r = ˆ η x β ˆ. The istribution of the istrict level resiuals (left sie plots) an q-q plots (right sie plots) are shown in Figure 1. The Figure 1 clearly reveals that the ranomly istribute istrict level resiuals an the line of fit oes not significantly iffer from the line y=0 as expecte in all the plots. The q-q plots also confirm the normality assumption. Therefore the moel iagnostics are fully satisfie for the ata. To valiate the reliability of the moel-base small area estimates we use the bias iagnostics, coefficient of variation (CV) an compute the 95 percent confience intervals. The bias iagnostics are use to investigate if the moel-base estimates are less extreme as compare to the irect survey estimates, when they are available, see [5]. The bias scatter plot of the moel-base estimates against the irect estimates is set out in Figure 2. The plot show that the moel-base estimates are less extreme as compare to the irect estimates, emonstrating the typical SAE outcome of shrinking more extreme values towars the average. We compute the CV to assess the improve precision of the moel-base estimates compare to the irect estimates. The CVs show the sampling variability as a percentage of the estimate. Estimates with large CVs are consiere unreliable (i.e. smaller is better). There are no internationally accepte tables available that allow us to juge how large is too large ([1] an [5]). Figure 3 presents the istrict-wise istribution of the percentage CV of moel base estimates an irect estimates. The estimate CVs show that moel-base estimates have a higher egree of reliability as compare to the irect estimates. In Table 2 we present the istricts-wise 95 percent confience intervals of the moel-base an the irect estimates. The stanar errors of the irect estimates are too large an therefore the estimates are unreliable. Note that for many istricts we can even not prouce the confience intervals ue to unavailability of stanar errors. The small area estimates iagnostic measures clearly epict that the moel-base estimates are reliable an more stable than the corresponing irect estimates (Figure 3). Table 2 presents the irect estimates an moel-base estimates along with 95 per cent confience intervals for the State of Uttar Praesh. These results show the egree of inequality with respect to istribution of poor househols in ifferent istricts. A critical review of Table 2 shows that in many istricts the lower boun (Lower) of 95% confience interval (CI) is negative which results in practically impossible an inamissible values of CI for irect estimates. In contrast, the moel estimate with precise CI an reasonable CV percent are reliable. This problem was mostly observe when there was no variability in the sample ata of istrict. For example
Int. Statistical Inst.: Proc. 58th Worl Statistical Congress, 2011, Dublin (Session CPS039) p.6572 where all y values in sample were 0 estimate irect proportions was 0. In such circumstance, SAE plays an important role in generating micro level statistics. The results clearly show the avantage of using SAE technique to cope up the small sample size problem in proucing the estimates or reliable confience intervals. These estimates can efinitely be useful for resource allocation an policy ecision-making relating the living conition of people in rural areas. 5. CONCLUSIONS The metho of estimation of proportions for small areas is well evelope ([6 an 9]), however, there is limite application in the area of agricultural or social sciences. Further, there is very less known application to the Inian ata, particularly, NSSO ata. In this article we emonstrate the application of SAE techniques to estimate the istrict level statistics of poor househols using survey an census ata. The iagnostic proceures clearly confirm that the moel-base istrict level estimates have reasonably goo precision. As the quantum of work involve in the conuct of Census is quite appreciable, Censuses are generally carrie out after a fixe perio of time. Thus, the Census ata is available only after a certain time perio. The NSSO survey, on the other han, contributes to proviing estimates on a regular basis at the State an National level. They o not provie sub-state level statistics. However, it is known that regional an national estimates usually mask variations (heterogeneity) at the sub-state or istrict level an rener little information for micro level planning an allocation of resources. REFERENCES [1] R. Ambler, D. Caplan, R. Chambers, M. Kovacevic, an S. Wang, Combining unemployment benefits ata an LFS ata to estimate ILO unemployment for small areas: an application of a moifie Fay-Herriot metho. Proceeings of the Int. Assoc. of Survey Stat., Meeting of the ISI, Seoul, August 2001. [2] G. E. Battese, R. M. Harter, an W. A. Fuller, An error component moel for preiction of county crop areas using survey an satellite ata, J. of the Amer. Stat. Assoc. 83 (1988), 28-36. [3] N. E. Breslow an D. G. Clayton, Approximate inference in generalize linear mixe moels, J. of the Amer. Stat. Assoc. 88(1993), 9-25. [4] R. E. Fay an R. A. Herriot, Estimation of income from small places: an application of james-stein proceures to census ata, J. of the Amer. Stat. Assoc. 74(1979), pp. 269-277. [5] F.A. Johnson, H. Chanra, J. J. Brown, an S. Pamaas, District-level estimates of institutional births in ghana: application of small area estimation technique using census an DHS ata, J. of Off. Stat. (2009), to appear. [6] G.W. Manteiga, M.J. Lombarìa, I. Molina, D. Morales, an L. Santamarìa, Estimation of the mean square error of preictors of small area linear parameters uner a logistic mixe moel, Comput. Stat. & Data Anal. 5, 2720-2733. [7] D. Pfeffermann, Small area estimation: new evelopments an irections. Int. Stat. Rev. 70(2002), pp.125-143. [8] J.N.K. Rao, Small Area Estimation. Wiley Series in Survey Methoology, John Wiley an Sons Inc, 2003. [9] A. Saei an R. Chambers, Small area estimation uner linear an generalize linear mixe moels with time an area effects, W.P. No. M03/15(2003), S3RI, University of Southampton, UK. [10] C.E. Särnal, B. Swensson, an J. Wretman, Moel Assiste Survey Sampling. Springer-Verlag, New York, 1992.